( ( )) y k af b x h - = - ( ( )) y af b x h k =

Pre-Calculus 12A
Combining Transformations
Focus on: i) sketching the graph of a transformed function by applying translations, reflections and stretches
ii) writing the equations of a function that has been transformed from the function y = f(x)
Multiple transformations can be applied to a function using the general transformation formula:
y  k  af (b( x  h))
OR
y  af (b( x  h))  k
***IMPORTANT to REMEMBER***
>>Stretches and Reflections (values of a and b) are performed first
>>translations (h-value and k-value) are performed second
Recall….
 a corresponds to the VERTICAL STRETCH about the x-axis by a factor of |a|
If a<o, then function is reflected in the x-axis
 b corresponds to the HORIZONTAL STRETCH about the y-axis by a factor of |
1
|
b
If b<0, then function is reflected in the y-axis
 h corresponds to a HORIZONTAL TRANSLATION
 k corresponds to a VERTICAL TRANSLATION
 x-intercepts are not affected by a vertical stretch
 y-intercepts are not affected by a horizontal stretch
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Pre-Calculus 12A
Example #1: Graph a Transformed Function
Describe the combination of transformations that must be applied to the function y = f(x) to obtain the
transformed function. Sketch the graph, showing each step of the transformation.
a) y = 3f(2x)
b) y = f(2x – 4)
y = f(x)
Solution:
a) Compare y = 3f(2x) to
y = af(b(x – h) + k
a = _____
b= _____
h = _____
k = _____
First, apply the vertically stretched about the x-axis by a factor of ________ .
Multiply the y-values by _______.
(2, 0) → (2,
)
(3, -1) → (3,
)
(6, -2) → (6,
)
(11, -3) → (11, )
Plot points and graph y = 3f(x).
Then, the new function is horizontally stretched about the y-axis by a factor of __________.
Using the new image points from above, multiply the x-values by ________.
(2, 0) → (
, 0)
(3, -3) → ( , -3)
(6, -6) → ( , -6)
(11, -9) →( , -9)
Plot points and graph y = 3f(2x).
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Pre-Calculus 12A
b) Sketch the graph of the transformed
function, showing each step of the
transformation:
y = f(2x + 4)
y = f(x)
Solution:
Rewrite the function into the form
y  af (b( x  h))  k
So, y = f(2x + 4) becomes:
y = f(2(x + 2))
a = ______
b = _____
h = _____
k = _____
Similar to the last example, apply the stretches and reflections first:
So, apply a horizontal stretch about the y-axis by a factor of _______. Multiply the x-values by this factor.
(2, 0) → (
, 0)
(3, -1) → (
, -1)
(6, -2) → (
, -2)
(11, -3) → (
, -3)
Plot the points and graph y = f(2(x)).
Next, apply the horizontal translation (h), moving graph 2 units to the _____________.
(1, 0) → (
(
, 0)
3
, -1) → ( , -1)
2
(3, -2) → (
(5.5, -3) → (
,-2)
, -3)
Plot these points and graph y = f(2x+4).
Page 3
Pre-Calculus 12A
Example 2: Combination of Transformations
Show the combination of transformations that should be applied to the graph of the function f(x) = x2 in order to obtain
the graph of the transformed function: g(x) = -2f(
Solution:
For g(x) = -2f(
1
(x – 1)) + 4 , a = _____,
2
Description
Horizontal stretch
about the y-axis by
a factor of 2
y=(
1 2
x)
2
1
(x – 1)) + 4.
2
b = ______,
h = ______,
k = ______
Mapping
( 2, 4)  (____, 4)
( 1,1)  (_____,1)
Graph
f(x)=x2
(0, 0)  (____, 0)
(1,1)  (____,1)
(2, 4)  (____, 4)
(x , y )  (____, y )
Vertical stretch
about the x-axis by
a factor of 2
y = 2(
1 2
x)
2
( 4, 4)  ( 4, ___)
( 2,1)  ( 2, ____)
(0, 0)  (0, ___)
(2,1)  (2, ____)
(4, 4)  (4, ___)
y=
1 2
x
2
(2x , y )  (2x , ___ y )
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Pre-Calculus 12A
Reflection in the xaxis
1 2
y = -2( x )
2
( 4,8)  ( 4, ___)
( 2, 2)  ( 2, ___)
(0, 0)  (0, ___)
(2, 2)  (2, ___)
(4,8)  (4, ___)
(2x , 2y )  (2x , __ y )
Translation of 1 unit
right and 4 units up
( 4, 8)  (___, ___)
1
y  2 f ( ( x  1)  4)
2
(0, 0)  (___, ___)
( 2, 2)  (___, ___)
(2, 2)  (___, ___)
(4, 8)  (___, ___)
(2x , 2y )  (_____, ____)
Page 5
Pre-Calculus 12A
Example 3: Write the Equation of a Transformed Function Graph
The graph of the function y = g(x) represents a transformation of the graph y = f(x). Determine the equation
of g(x) in the form y = a f (b(x – h)) + k. Explain your answer.
Solution:
Locate key points on f(x) and their image
points on g(x):
(-1, 1) >> (1, -7)
(0, 0) >> (3, -4)
(1 , 1) >> (5, -7)
The point (0, 0) is not affected
by stretches or reflections so we
can use this to determine the
horizontal and vertical translations.
So, this point has moved:
_____ units right
y  g ( x)
_____ units down
h = ______ k = ______
To determine horizontal and vertical stretch, compare distances between key points.
Horizontally, keypoints are _____ units apart on f(x) and ______ units apart on g(x). So, multiplied by
a factor of _____.
Vertically, keypoints are ____ unit apart on f(x) and ___ units apart on g(x). So, multiplied by a factor
of _____.
Also, we can see the graph has been reflected in the x-axis so a <0.
So a= _________
and b = _____________.
y  af (b( x  h))  k
1
Solution: y  3 f ( ( x  3))  4
2
Put it all together in the form:
Page 6

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